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Numerical mathematics is a subtopic of scientific computing. The focus lies on the efficiency of algorithms, i.e. speed, reliability, and robustness. This leads to adaptive algorithms. The theoretical derivation und analyses of algorithms are kept as elementary as possible in this book the needed sligtly advanced mathematical theory is summarized in the appendix. Numerous figures and illustrating examples explain the complex data, as non-trivial examples serve problems from nanotechnology, chirurgy, and physiology. The book addresses students as well as practitioners in mathematics, natural sciences, and engineering. It is designed as a textbook but also suitable for self study